9 Conclusion

This article provides a formalization of Hierarchical Task Network
planning that, unlike the UMCP formalism [Erol, Nau, Hendler, 1994b],
includes actions with temporal extent. We introduce a sound and
complete algorithm that can be used to generate a plan, coordinate a
group of agents with hierarchical plans, and interleave planning and
coordination.

The algorithms for summarizing propositional state and metric resource
conditions and effects at abstract levels and the mechanisms that
reason about this summary information can facilitate the construction
of other planning and coordination systems that reason about plans at
multiple levels of abstraction. These mechanisms for reasoning about summary information determine whether a
task (at any level of abstraction) must or may achieve, clobber, or
undo a condition of another task under partial order constraints on
endpoints of tasks. Built on these mechanisms, other mechanisms determine
whether a group of agents can decompose and execute a set of partially
ordered abstract tasks in any way (), might decompose and
execute them in some way (), or cannot execute them
consistently in any way ().

These algorithms enable a planning system to find solutions at
multiple levels of abstraction without needing to fully detail the
task hierarchy. These abstract solutions support flexible execution
by remaining uncommitted about which of the alternative methods will
be selected at runtime, based on the circumstances, to achieve plan
subgoals.

Our complexity analyses and experiments in different problem domains
have quantified the benefits of using summary information for a
refinement planning and local search scheduling algorithm. There is a
potential doubly exponential speedup of
for ways to resolve a
conflict, a hierarchy branching factor , a
depth of the hierarchy , and an abstract solution depth . An
exponential speedup is obtained if abstract solutions are found, if
there are fewer summary conditions at abstract levels, or if alternative
decomposition choices lead to varying numbers of threats.
These conditions for exponential improvement are a
significant relaxation compared to prior work, and the performance improvement is greater.

A domain modeler can run the summarization algorithms offline for a
library of plan hierarchies so that summary information is available
for the coordination and planning of any set of goal tasks supported
by the library. Using algorithms for reasoning about summary
information, agents can discover with whom they should coordinate and
over which states and resources they must coordinate/negotiate.
Communicating summary information at
different levels of abstraction reduces communication costs exponentially
under conditions similar to those reducing computation time.

The use of summary information in a local search planner (like ASPEN, Section 6.3) is another
contribution of this work. The strength of local search algorithms
is their ability to efficiently reason about large numbers of tasks
with constraints on metric resources, state variables, and other
complex resource classes. By integrating algorithms for reasoning
about summarized propositional state and metric resource constraints
into a heuristic local search
planner/scheduler, we enable such scalable planning systems to scale to even
larger problem domains.
This use of summary information in a different
style of planner demonstrates the applicability of abstract reasoning
in improving the performance of different kinds of planning (and plan
coordination) systems.

Future work is needed to evaluate the use of summary information in
other planning and scheduling systems and for wider classes of
problems requiring more expressive representations for resources and
temporal constraints. Already, an approach for exploiting cooperative action
among agents based on summary information has been developed [Cox, Durfee, 2003]. Other promising approaches include abstracting
other plan information, such as probabilistic conditions and effects
and classes of resources and states (e.g. location regions and
sub-regions). More work is also needed to understand how and when to
communicate summary information in a distributed planning system.

The authors wish to thank Pradeep Pappachan, Gregg Rabideau, and Russell Knight for help with implementation.
We also thank our anonymous reviewers for
their many valuable suggestions.
This work was performed at the Jet Propulsion Laboratory,
California Institute of Technology, under contract with the National
Aeronautics and Space Administration, and at the University of Michigan
supported in part by DARPA (F30602-98-2-0142).